ABSTRACT
This chapter is devoted to bilinear forms. We previously defined the concept of an m-multilinear map from vector spaces V1, . . . , Vm to the vector space W. A particularly important special case is when m = 2. Such functions were referred to as bilinear maps. Bilinear maps are important because of their role in the definition of the tensor product of two spaces, which is the subject of chapter ten. Bilinear forms (bilinear maps to F, the underlying field) arise throughout mathematics, in fields ranging from differential geometry and mathematical physics on the one hand, to group theory and number theory on the other. In the introductory section of this chapter we develop some basic properties of bilinear maps and forms, introduce the notion of a reflexive form, and prove that any reflexive form is either alternating or symmetric. The second section is devoted to the structure of symplectic space, a vector space equipped with an alternating form. In the third section, we define the notion of a quadratic form and develop the general theory of an orthogonal space. In particular, we proveWitt’s theorem for an orthogonal space when the characteristic of the field is not two. The fourth section deals with orthogonal space over a perfect field of characteristic two. Finally, section five is concerned with real orthogonal spaces.
